Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Vera Serganova
University of California at Berkeley
Borel-Weil-Bott theorem and Bernstein-Gelfand-Gelfand reciprocity for classical supergroups
In 2003 J. Brundan found remarkable connections between the category F of finite-dimensional representations
of GL(m,n) and tensor representations of GL(∞). In the recent paper Brundan and Stroppel
develop this idea further using a categorification approach and construct a certain Koszul algebra which
completely describes the structure of the category F. The important ingredient of this approach is the
Bernstein-Gelfand-Gelfand reciprocity law which relates projective and simple objects in F via so called
Kac modules. The characters of Kac modules are easy to compute, hence knowing of multiplicities of Kac
modules in projective modules allows one to obtain the character of a simple module. Unfortunately, Kac
modules do not exist for other classical supergroups. We will show that BGG reciprocity holds if in place
of a Kac module one takes a virtual module given by the Euler characteristic of a line bundle on a flag
supermanifold. We give a combinatorial algorithm for decomposition of a projective module into a sum of
these virtual modules for the orthosymplectic supergroup and explore connections with representations
of GL(∞) on this case. This talk is based on a joint work with C. Gruson.
Catharina Stroppel
Universität Bonn
t.b.a.
Joris Van der Jeugt
Gent University
Wigner quantization and representations of Lie superalgebras
For many quantum systems described by a Hamiltonian involving oscillators or oscillator-like interactions,
the technique of Wigner quantization leads to algebraic compatibility conditions between operators. These
conditions can be solved in terms of Lie superalgebra generators, in particular in terms of the Lie superalgebras
osp(1|2n) and gl(1|n). This implies that the (unitary) representations of these Lie superalgebras
are important in order to describe some physical quantities of these systems (position spectrum, energy,
angular momentum contents).
We shall focus on some examples where interesting representation theoretic questions appear. For
osp(1|2n) this involves the characters (and a character formula) of a class of infinite-dimensional lowest
weight representations, and the decomposition of these representations with respect to (physically relevant)
subalgebras. For gl(1|n), this involves finite-dimensional representations (with known characters,
e.g. in terms of Schur functions), and here we present some generating function techniques to describe
decompositions with respect to subalgebras. In doing so, we use various computational results from the
field of symmetric and supersymmetric Schur functions.
Weiqiang Wang
University of Virginia
A super duality approach to the representation theory of Lie superalgebras
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